A diverging Mirror form an image of height 2.4cm if the focal length of the mirror is 20 cm find the height and distance of the object from the front of the mirror
To find the height and distance of the object from the front of the mirror, we can use the mirror formula:
1/f = 1/d_o + 1/d_i
Where:
f = focal length of the mirror
d_o = distance of the object from the mirror
d_i = distance of the image from the mirror
Given:
f = 20 cm
d_o = ?
d_i = ?
Since it is a diverging mirror, the focal length will be negative (-20 cm).
By rearranging the formula, we can solve for d_o:
1/d_o = 1/f - 1/d_i
Substituting the given values:
1/d_o = 1/-20 - 1/d_i
Now, let's calculate the distance of the image from the mirror (d_i):
Using the mirror magnification formula:
m = -d_i/d_o
Given that the height of the image (h_i) is 2.4 cm, and the height of the object (h_o) is unknown, we can use the magnification formula to find the ratio of the heights:
m = h_i/h_o
Substituting the given values:
2.4/h_o = -d_i/d_o
To solve for d_i, we can rearrange and solve the equation for d_o:
d_i = (-h_i/h_o) * d_o
Now, we can substitute this value of d_i back into our original equation:
1/d_o = 1/-20 - 1/((-h_i/h_o) * d_o)
Now, simplify the equation by combining terms:
1/d_o + 1/((-h_i/h_o) * d_o) = 1/-20
To make it easier, let's substitute a = 1/d_o, b = -h_i/h_o, and c = 1/-20:
a + 1/(b * d_o) = c
Now, multiply both sides of the equation by (b * d_o) to get rid of the denominator on the left side:
a(b * d_o) + 1 = c(b * d_o)
Expand and simplify:
a * b * d_o + 1 = c * b * d_o
Rearrange the equation to solve for d_o:
a * b * d_o - c * b * d_o = -1
Factor out d_o:
d_o * (a * b - c * b) = -1
Divide both sides by (a * b - c * b):
d_o = -1 / (a * b - c * b)
Substituting the values of a, b, and c:
d_o = -1 / (1/d_o * (-h_i/h_o) - 1/-20)
Now, we can solve for d_o by substituting the value of d_o on the right side with the obtained expression:
d_o = -1 / (1/((-1 / (1/(-20 - 1/(-h_i/h_o)) * (-h_i/h_o))
Simplifying further:
d_o = -1 / (1/((-1 / (1/(-20 + h_i/h_o)) * (-h_i/h_o))
Now, we can substitute the given values of h_i = 2.4 cm and f = 20 cm into the equation and calculate d_o.